Gromov-Witten theory of toroidal orbifolds and GIT wall-crossing

Toroidal 3-orbifolds \((S^1)^6/G\), for \(G\) a finite group, were some of the earliest examples of Calabi-Yau 3-orbifolds to be studied in string theory. While much mathematical progress towards the predictions of string theory has been made in the meantime, most of it has dealt with hypersurfaces in toric varieties. As a result, very little is known about curve-counting theories on toroidal orbifolds. In this paper, we initiate a program to study mirror symmetry and the Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence for toroidal orbifolds. We focus on the simplest example \([E^3/\mu_3],\) where \(E\subseteq\mathbb{P}^2\) is the elliptic curve \(\mathbb{V}(x_0^3+x_1^3+x_2^3).\) We study this orbifold from the point of GIT wall-crossing using the gauged linear sigma model, a collection of moduli spaces generalizing spaces of stable maps. Our main result is a mirror symmetry theorem that applies simultaneously to the different GIT chambers. Using this, we analyze wall-crossing behavior to obtain an LG/CY correspondence relating the genus-zero Gromov-Witten invariants of \([E^3/\mu_3]\) to generalized Fan-Jarvis-Ruan-Witten invariants.